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UC-NRLF 


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IN   MEMORIAM 
FLOR1AN  CAJOR1 


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1SK 
c_ 

l~73 


ADDRESS 


ON   THE 


STUDY  OF  MATHEMATICS, 


BY 


erastus\everett,  a.m., 

Principal  of  the  ORLEANS  HIGH  SCHOOL,  New  Orleans, 


AT  THE  CLOSE  OF  THE  ANNUAL  EXAMINATION,  AUGUST  18, 1852. 


NEW    ORLEANS: 
Printed  by  Hinton  &  Bro.,  102  Gravier  street. 

1852, 


CAJ0R1 


QA7 
e2 


ADDRESS 


Ladies  and  Gentlemen  : 

It  becomes  my  agreeable  duty  to  thank  you  for 
the  interest  which  you  have  taken  in  the  exercises  of 
this  day.  We  were  all  once  children,  and  we  shall 
never  forget  how  much  the  approbation  of  our  seniors 
encouraged  us  in  the  career  which  we  were  just  begin- 
ning, and  how  we  then  ventured  for  the  first  time  to 
aspire  to  honors  which  merit  only  can  win.  I  thank 
you,  not  only  in  my  own  behalf  and  that  of  the  young 
gentlemen  of  the  Institution  who  have  appeared  before 
you,  but  in  behalf  of  my  learned  colleagues.  They  have 
faithfully  and  ably  seconded  me  in  my  humble  endea- 
vors to  impart  to  the  students  entrusted  to  my  care, 
an  accurate  and  extended  knowledge  of  those  branches 
of  learning  which  will  be  found  indispensable  to  their 
success  and  honorable  standing  in  future  life.  And 
to  you,  young  gentlemen,  let  me  recommend  not  to 
forget  the  applause  you  have  this  day  received.  You 
perceive  that  your  career  is  watched  with  solicitude  by 
your  parents,  and,  though  the  love  which  God  has 
implanted  in  their  bosoms  would  never  be  quite  extin- 
guished by  your  obscurity,  it  is  increased  ten  fold  and 
there  is  superadded  to  it  an  honest  pride  which  parents 
only  can  feel,  when  they  see  that  you  will  not  only 
transmit  their  names  to  the  next  generation,  but  that 


the  honor  of  these  names  which  they  have  rendered 
illustrious  shall  still  be  guarded  when  they  shall  sleep 
with  their  fathers,  by  you  their  sons,  their  joy,  their 
pride,and,  it  maybe,  the  hope  of  their  declining  years. 
In  accordance  with  ancient  custom  in  Literary  In- 
stitutions, we  close  our  Annual  Examination  and  our 
Academic  Year  with  dramatic  and  oratorical  exercises. 
We  ought,  however,  in  order  that    we  may  not  be 
misunderstood,  to  remark,  that  in  our  daily  instruc- 
tions, we  attach  to  these  exercises  a  secondary  impor- 
tance.    The  useful  rather  than  the  agreeable  occupies 
our  chief  attention,  and  the  great  secret  of  teaching  is 
to  invest  the  useful  with  a  charm  which  shall  cause  the 
student  to  pursue  it  with  eagerness  and  delight.     We 
ever  bear  in  mind  the  fact  that  those  youths  will  soon 
be  men ;  that  they  will  be  called  upon  to  perform  the 
responsible  duties  of  life ;  and  that  in  the  performance 
of  those  duties,  accurate  knowledge  will  fit  them  for  all 
emergencies,  while  the  mere  graces  of  elocution,  with- 
out this  knowledge,  would  only  make  their  ignorance 
the  more  conspicuous*  and  expose  to  contempt  such  as 
might  have  dwelt  in  peaceful  obscurity.     In  making 
these  remarks*  we  must  not  be  understood  as  underva- 
luing the  power  of  true  eloquence;  but  it  is  vain  to 
cultivate  this*  to.  any  great  degree,  till  we.  have  first 
subjected  the  mind  to  rigid  discipline,  and  stored  it 
with  useful  learning  by  years,  of  arduous  study.     The 
caution  of  Quintilian,  with  regard  to  style,  should  be 
adopted  as  the  motto  of  every  teacher :  "  Curam  verbor 
rum*,  rerum  volo  esse  solicitudinem,"  says  this  great 
^fester. 

Among  the  branches,  of  learning  to  which  we  attach 


a  primary  importance,  are  Mathematics  and  the  Eng- 
lish, the  French,  and  the  Spanish  language.  Ours  is 
essentially  a  commercial  community,  and  the  young 
men  now  in  school  will  soon  have  to  take  a  part  in 
conducting  the  mercantile  transactions  of  the  world.  It 
appears  to  us  that  this  fact  has  too  often  been  lost 
sight  of,  and  that  these  branches,  and  particularly 
Mathematics,  have  not  received  that  attention  which 
their  importance  deserves.  Fully  impressed  with  the 
truth  of  this  remark,  we  have  endeavoured  of  late  to 
raise  the  standard  of  merit  in  this  department,  and  to 
impart  a  thorough  knowledge  of  Mathematics  and  to 
elucidate  its  principles  from  its  simplest  elements  to 
its,  most  abstruse  truths  and  most  remote  conclusions. 

I  have  thought  that  I  could  not  choose  a  more  impor- 
tant theme,  analogous  to  the  occasion,  than  the  study 
of  this  science.  I  am  aware  that  some  turn  away  from 
this  subject  as  void  of  interest,  but  I  have  no  fears  of 
meeting  with  any  such  in  the  polite  and  intellectual 
assembly  whom  it  is  now  my  pleasure  to  address.  And 
you  will,  I  doubt  not,  accompany  me  while  we  devote 
a  short  time  to  the  consideration  of  this  profitable  and 
interesting  subject. 

The  Ancients  had  but  a  limited  knowledge  of  this 
science.  The  knowledge  of  the  Egyptians,  which  was 
probably  derived  mostly  from  the  Persians  and  the 
Hindoos,  was  confined  almost  exclusively  to  the  ele- 
ments of  geometry.  They  applied  this,  however,  to 
mechanics  so  far  as  to  be  able,  by  the  application  of 
them  to  huge  engines,  to  rear  those  massive  structures 
which  are  the  wonder  of  the  world.  The  Grecians, 
particularly  Thales  and  Pythagoras,  but,  above  all, 


Euclid,  carried  geometry  to  such  perfection  that  little 
remained  to  succeeding  ages  but  to  study  the  admirable 
analyses  which  they  left  behind  them.  But  arithmetic, 
algebra  and  trigonometry  were  quite  unknown  to  them. 
These  sciences  may  be  said  to  have  had  their  birth 
among  the  Saracens,  who  have  done  more  than  any 
other  people  to  extend  the  province  of  Mathematics. 

The  mission  of  the  Saracens  as  propagators  of  learn- 
ing, began  about  the  middle  of  the  eighth  century; 
and  into  every  part  of  their  then  extended  empire  they 
infused  the  love  of  learning  that  they  owed  to  the 
Grecian  authors  with  whom  they  then  for  the  first 
time  became  acquainted  by  translations  into  the 
Arabic;  and  this  love  of  learning,  which  with  the 
Greeks  was  the  offspring  of  reason,  became  with 
the  Saracens  a  burning  zeal,  an  inextinguishable  en- 
thusiasm. They  carried  the  cultivation  of  letters  with 
them  in  all  their  conquests,  and  at  a  period  in  which 
all  the  world  was  buried  in  darkness,  and  barbarism 
had  taken  the  place  of  civilization,  Science  took  refuge 
in  the  court  of  the  Caliphs  of  Bagdad.  Here  she  found 
protection,  and  here  she  made  some  of  her  most  bril- 
liant conquests.  Not  only  did  the  people  of  this  won- 
derful empire  preserve  to  the  world  the  learning  which 
they  had  derived  from  Greece,  but  they  enriched  it 
with  immense  additions  by  their  own  researches. — 
Among  them  chemistry  had  its  origin.  And, 
though  the  glory  of  the  most  brilliant  discoveries 
in  this  science  was  reserved  for  the  English  and 
Erench  savans  of  our  own  days,  the  Saracens  made 
some  discoveries  which  are  valuable.  They  first  dis- 
covered by  distillation,  pure  alcohol,  a  chemical  agent 


whose  value  can  scarcely  be  overrated.  By  the  invention 
of  gunpowder  they  prepared  the  way  for  revolutionising 
the  ancient  mode  of  warfare  and  substituting  one  less 
destructive  of  human  life.  That  they  made  many 
contributions  to  astronomy,  is  attested,  not  only  by  his- 
tory, but  by  the  names  of  numerous  stars  whose  Arabic 
prefixes  show  that  their  places  were  either  first  noted 
by  the  Saracens  or  that  they  first  made  use  of  these  stars 
as  bases  of  astronomical  calculations.  The  former  is 
the  case  with  many  stars  of  the  lesser  magnitudes  and 
the  latter  with  such  stars  as  Aldebaran,  one  of  the 
Hyades  in  the  constellation  Taurus,  and  Alpheritz  and 
Almaach  in  Andromeda.  The  heavens  are  "sowed  with 
stars,"  "thick  as  a  field,"  that  first  received  their  names 
from  the  Saracens.  They  have  thus  built  monuments 
for  themselves  in  the  firmament  "more  durable  than 
brass."  But  their  greatest  triumphs  were  in  the  science 
of  algebra.  True,  we  are  told  they  were  not  the  in- 
ventors of  this  science,  though  its  etymology  is  a  strong 
presumption  in  favor  of  this  supposition,  but  certain  it 
is,  they  were  the  first  who  introduced  it  into  Europe. 
This  was  a  gigantic  step  in  the  career  of  Mathematics. 
Algebra,  in  the  hands  of  Descartes,  Newton  and  La- 
7  place  leid  to  those  astonishing  results  in  astronomy 
which  illustrate  the  last  century  and  the  present.  Be- 
sides this,  to  the  same  ingenious  people,  we  owe  the 
invention  of  the  decimal  system.  From  the  time  of 
this  invention  arithmetic  first  took  its  place  among 
the  sciences  as  an  important  branch  of  Mathema- 
tics. To  the  Ancients  arithmetic  was  almost  wholly 
unknown;  and  when  we  consider  the  difficulty  of  cal- 
culating by  Roman  letters  we  shall  cease  to  wonder 


8 

that  the  arithmetical  knowledge  of  the  Ancients  was 
quite  elementary.  The  art  of  book-keeping,  since  in- 
vented bj  the  Italians,  would  be  quite  impracticable 
without  the  decimal  system.  We  are  thus  prepared  to 
acknowledge  the  debt  of  gratitude  which  the  commer- 
cial world  owe  to  the  Saracens. 

Valuable  additions  have  been  made  to  arithmetic, 
algebra  and  trigonometry  within  the  last  two  centuries. 
Arithmetic  is  receiving  improvements  almost  every 
year,  and  America  may  justly  claim  the  honor  of  hav- 
ing furnished  some  admirable  treatises  on  this  subject. 
Algebra  has  been  greatly  improved  in  the  hands  of  the 
French,  the  English  and  the  Germans.  The  method  of 
differential  calculus  was  discovered  by  Newton  as  early 
as  1669,  and  a  few  years  after  by  Leibnitz.  Such  new  ap- 
plications of  the  binomial  theorem  were  made  by  Newton 
as,  joined  with  his  other  contributions  to  this  science, 
have  placed  him  above  all  other  mathematicians  both 
ancient  and  modern.  Logarithmic  tables,  which  shor- 
ten immeasurably  the  former  tedious  operations  in 
trigonometry,  were  published  by  Lord  Napier  in  1614. 

Having  given  a  sketch  of  the  history  of  this  science, 
let  us  consider  some  of  its  advantages.  Anc^  first,  it 
is  a  universal  language.  The  proposition,  for  ekample, 
that  the  sum  of  two  quantities  multiplied  hy  their 
difference  is  equal  to  the  difference  of  their  squares,  is 
enunciated  by  means  of  a  few  characters  extremely 
simple,  known  to  all  algebraists  of  every  tongue  all  over 
the  world.  The  same  may  be  said  of  every  propo- 
sition. 

But,  second,  not  only  is  this  a  universal  language, 
but,  what  is  of  far  greater  moment,  it  has  to  do  with 


9 

i 
general  truths.     It  thus  becomes  the  most  beautiful 

example  of  pure  intellection  that  can  occupy  the 
human  mind.  It  seems  to  be  the  nearest  approach 
that  the  mere  intellect  of  man  can  make  to  the 
mind  of  the  Infinite.  The  hypotheses  of  metaphy- 
sicians are  nothing  to  it.  They  rest  upon  the  shadowy 
foundations  of  theory,  this  upon  the  immovable  foun- 
dations of  demonstrated  truth.  The  hypotheses  of  the 
metaphysician  of  yesterday  are  overthrown  by  the  me- 
taphysician of  to-day,  that  he  may  substitute  other 
hypotheses  to  be  overthrown  in  their  turn  to-morrow. 
But  the  demonstrations  of  the  mathematician  will  sur- 
vive "the  wreck  of  matter  and  the  crush  of  worlds." 
Hence,  there  is  a  satisfaction  in  mathematical  demon- 
stration which  the  merely  moral  reasoner  cannot  con- 
ceive of.  Pythagoras  is  said  to  have  sacrificed  a 
hecatomb  of  oxen  when  he  discovered  that  the  square 
of  the  hypothenuse  is  equal  to  the  squares  of  the  oppo- 
site sides.  Archimedes  discovered  a  new  law  in  hydro- 
statics and,  smit  with  divine  enthusiasm,  he  ran  through 
the  streets  of  his  native  city  shouting  to  all  he  met,  "  I 
have  found  it !  I  have  found  it !"  Tradition  relates  that 
this  devotee  to  the  science  of  which  we  are  treating 
fell  a  sacrifice  to  his  enthusiasm.  During  the  storm- 
ing of  Syracuse,  he  was  sitting  at  the  market-place 
considering  the  properties  of  the  circle  by  means  of 
a  diagram  which  he  had  made  in  the  sand,  and  when 
a  Roman  soldier  approached,  his  entreaty  was  not 
"save  my  life,"  but  "spare  my  circle."  The  ruthless 
soldier,  heedless  of  the  man  of  science,  struck  him 
down.  Nor  is  there  anything  improbable  in  these  tra- 
ditions. Well  attested  examples  have  not  been  wanting 


10 

in  modern  times  of  men  who  held  science  dearer  than 
life.  The  fate  of  the  chemist  Lavoisier  is  probably 
known  to  most  of  us.  He  fell  a  sacrifice  to  the  vio- 
lence of  the  times  during  the  Reign  of  Terror  in 
1794,  and  when  condemned  to  death,  he  plead  for  a 
reprieve  of  a  few  days  till  he  should  have  time  to 
complete  some  experiments  upon  which  he  was  then 
engaged.  The  answer  of  his  judges  was,  "  The  Re- 
public has  no  need  of  chemists ;"  and  he  was  compelled 
to  leave  his  retorts  upon  the  fire  and  go  by  the  same 
road  as  Chenier,  and  Malesherbes,  and  Madame  Roland, 
to  that  world  where  "there  is  no  work,  nor  device,  nor 
knowledge,  nor  wisdom." 

Third :  Another  obvious  advantage  of  the  study  of 
Mathematics  is  the  mental  discipline  which  it  imparts. 
It  does  this  in  two  ways  :  first,  it  fixes  the  attention ; 
second,  the  premises  being  certain,  the  mind  is  freed 
from  that  doubt  which  too  often  attends  moral  reason- 
ing. 

It  is  manifest  that  without  attention,  the  mind  can 
not  advance  a  single  step;  for,  if  an  intermediate  step 
is  wanting,  all  the  preceding  conclusions  are  of  no  avail. 
The  process  of  reasoning  is  a  chain,  and  if  one  link  is 
broken  the  entire  chain  is  broken. 

But  attention  would  be  of  no  avail  as  a  means  of 
mental  discipline  were  it  not  for  the  fact  that  the 
mind  being  free  from  uncertainty  with  regard  to  the 
point  of  departure  reasons  with  vigor  and  with  that 
confidence  in  its  own  conclusions  which  cannot  fail  to 
strengthen  the  powers  of  the  mind.  For,  let  it  be 
remarked  that  it  is  not  every  kind  of  mental  exercise 
that  serves  as  a  mental  discipline.     On  the  contrary, 


11 

when  we  reason  merely  for  the  sake  of  argument,  as 
in  case  of  a  debate  in  which  we  are  chosen  as  champions 
of  a  proposition  without  regard  to  its  merits,  or  when 
we  take  our  departure  from  doubtful  premises,  as  in 
points  of  scholastic  theology,  the  mind  instead  of  being 
invigorated  becomes  enfeebled.  So  far  have  some 
trifled  with  their  powers  of  debate,  that  they  have 
habitually  played  the  hypocrite  by  defending  what 
they  knew  to  be  wrong  merely  to  show  their  skill ;  and 
God  has  inflicted  upon  them  a  punishment  adequate  to 
their  offense.  He  has  filled  their  minds  with  universal 
doubt;  so  that,  having  begun  by  pretending  to  be 
sceptics,  they  end  in  becoming  such,  and  doubt  every- 
thing, even  the  existence  of  a  God,  nay,  even  their 
own  existence  and  the  existence  of  matter.  Our  mad- 
houses are  filled  with  that  other  class  who,  sincere  but 
too  bold,  have  striven  to  attain  objects  beyond  their 
reach,  and  have  been  overtaken  by  the  just  judgment 
of  Heaven  who  has  smitten  their  minds  with  hopeless 
impotency.  And  they  furnish  melancholly  examples 
of  the  apophthegm  of  Pope  : 

Aspiring  to  be  gods,  if  angels  fell, 
Aspiring  to  be  angels,  men  rebel. 
To  be  convinced  of  the  vast  superiority  which  math- 
ematical reasoning  possesses,  as  a  means  of  mental 
discipline,  over  all  methods  of  moral  reasoning, 
let  us  glance  at  the  peculiarities  of  the  two  most  re- 
markable of  these  methods. — Aristotle's  method 
consisted  in  discovering  a  new  truth  from  the 
relation  which  exists  between  two  known  truths. 
But  here  arises  a  difficulty  on  our  first  setting  out. 
What   are   known   truths?      What   one    receives  as 


12 

truth,  another  considers  as  either  doubtful  or  false.  We 
hope  we  shall  not  be  considered  pedantic  if  we  intro- 
duce, for  the  sake  of  illustration,  a  syllogism  in  form. 
Let  us  take  for  our  major  premise  the  proposition 
that  All  republics  are  good  governments ;  and  for 
our  minor  premise,  France  and  Mexico  are  repub- 
lics. The  conclusion  inevitably  follows  that  France 
and  Mexico  are  good  governments.  Now,  this  con- 
clusion is  erroneous  because  the  major  premise  is 
false.  It  is  not  true  that  all  republics  are  good 
governments.  The  same  doubt,  in  many  instances, 
attends  the  method  of  Lord  Bacon.  And  here  it 
is  proper  to  remark,  that,  with  respect  to  his  phi- 
losophy, there  has  been  a  very  general  misappre- 
hension. Because  the  philosophy  of  Aristotle  ac- 
complished little  and  that  of  Bacon  accomplished 
much,  it  has  been  concluded  that  the  philosophy  of 
Bacon  is  opposed  to  that  of  Aristotle.  But  the  truth 
is,  Bacon  came  to  Aristotle's  assistance.  He  only 
completed  what  his  predecessor  had  left  unfinished.  He 
did  but  consolidate  the  foundations  which  were  too 
weak  to  support  the  noble  structure  that  the  Stagy- 
rite  had  reared  upon  them.  Why  did  the  system  of 
Aristotle  fail  in  producing  those  results  which  followed 
the  introduction  of  Bacon's  method?  It  was  not  for 
want  of  acute  minds,  for  there  were  among  the  school- 
men minds  as  acute  as  that  of  Blaise  Pascal  or  Jona- 
than Edwards.  It  was  not  for  want  of  attention 
to  philosophical  subjects ;  for  the  schoolmen  attended 
to  nothing  else.  It  was  simply  because  the  Aristote- 
lians took  for  grantedwhat  had  never  been  proved, 
and  hence,  their  whole  system  was  but  a  tissue  of  argu- 


13 

tentative  romance.  They  reasoned  and  reasoned  till 
their  brains  were  turned,  upon  subjects  which  it  is 
utterly  impossible  to  decide  and  the  decision  of  which, 
even  though  it  were  possible,  would  be  of  no  use  either 
practically  or  philosophically.  And  why  all  this  ? 
Why,  simply  because  the  great  Master  had  left  his 
major  premise  unproved.  Bacon  taught  us  to  establish 
this  by  experiment.  It  is  for  that  reason  that  his  method 
has  been  called  the  inductive  method.  Though  not  the 
first,  as  has  been  remarked  by  the  ingenious  Mr.  Macau- 
lay,  to  make  use  of  this  method,  he  was  the  first  who  in- 
sisted on  it  as  essential  to  the  establishment  of  any  new 
truth.  Having  furnished  a  method  by  which  to  prove 
the  major  premise  of  Aristotle,  he  has  laid  the  syllogis- 
tic method  in  foundations  that  can  never  be  moved,  and 
hence  the  triumphs  of  his  system.  We  have,  then,  dis- 
tinctly before  us  the  peculiar  merit  of  both  these  great 
masters.  The  former,  taking  certain  truths  for  granted, 
pointed  out  the  process  of  ratiocination  by  which  to 
arrive  at  other  truths,  and  showed  that,  without  this 
process,  no  conclusion  could  be  satisfactory.  The  latter 
has  furnished  us  a  method  by  which  to  prove  what  the 
other  had  taken  for  granted,  and  has  supplied  the  link 
which  was  wanting  to  make  the  chain  of  reasoning 
complete.  Well  may  we  exclaim, ' '  Par  nobile  fratrum." 
You  have  done  all  that  human  genius  could  do  to  avoid 
error  and  to  arrive  at  truth.  But  you  have,  with  all 
your  wisdom,  failed  to  establish  many  truths  on  which 
depends  the  happiness  of  millions.  In  some  kinds  of 
moral  reasoning  your  methods  have  the  certainty  of 
mathematical  demonstration.  In  others,  your  united 
efforts  have  not  been  able  to  save  us  from  painful 


14 

doubt.     On  the  great  question  of  the  truth  of  Revela- 
tion, proved  as  it  is  by  the  united  testimony  of  disinter- 
ested witnesses,  and  by  an  immense  amount  of  internal 
evidence  consisting  of  the  Divine  nature  of  the  truths 
revealed,  the  fulfilment  of  the  prophecies  with  regard 
to  ancient  cities,  and  the  oriental  cast  of  the  scenery, 
imagery,  manners  and  customs,  with  which  the  Holy 
Scriptures  abound,  we  cannot  entertain  the  shadow  of  a 
doubt.     Indeed,  the  making  of  an  apocryphal  book  like 
the  Bible  would  be  a  greater  miracle  than  any  which  it 
records  in  proof  of  the  mission  of  the  Messiah.     And 
the  proofs  of  the  incidents  of  our  Saviour's  life  and 
death  are  as  well  established  as  those  of  the  life  and 
death  of  Julius  Caesar.     They  both  rest  upon  the  basis 
of  history.     Nor  must  it  be  supposed  that  all  truth 
is  to  be  doubted  because  it  cannot  be  proved  mathema- 
tically.    It  would  be  just  as  absurd  to  apply  the  pro- 
cess  of  algebraic  equations  to  prove  the  truth  of  a 
historical  fact  as  it  would  to  attempt  to  measure  Mont 
Blanc  by  means  of  a  syllogism.     The  two  systems  of 
proof  have  nothing  in  common  with  each  other.     But 
on  many  points  of  history  the  world  is  still  divided  in 
opinion.    The  question  is  not  yet  settled  whether  Mary 
Queen  of  Scots  was  guilty  of  the  murder  of  her  hus- 
band.    The  same  may  be  said  of  the  justice  of  the 
death  of  Charles  I,  the  justifiableness  of  the  Crusades, 
and  a  thousand  other  questions  which  your  own  famili- 
arity with  history  will  suggest  to  you.    So,  in  questions 
of  law  and  ethics.     Our  most  learned  jurisconsults  are 
retained  to  maintain  opposite  sides  of  disputed  points, 
and  so  difficult  is  it  to  arrive  at  truth,  that  the  deci- 
sions of  our  courts  are  often  inconsistent  with  each 


15 

other.  Questions  of  casuistry  encounter  us  everywhere, 
and  few  are  wiser  after  having  been  tossed  upon  the 
ocean  of  life  for  years  than  when  they  first  launched 
their  frail  bark  upon  the  stream  of  childhood.  Philoso- 
phers have  been  reasoning  about  fatality  and  free-will 
for  more  than  two  thousands  years,  and  it  is  doubtful 
whether  we  have  advanced  or  retrograded.  These  sub- 
jects are  involved  in  mystery  from  their  very  nature, 
and  any  attempts  to  elucidate  them  must  end  in  contra- 
diction and  obscurity.  Man's  will,  they  tell  us,  is  free, 
because  he  wills  what  he  chooses  to  will ;  but  they  tell  us 
further,  that  he  always  chooses  and  always  must  choose 
according  to  the  strongest  motive,  and  the  motive  is 
what  moves.  In  other  words,  causing  the  technicali- 
ties to  disappear,  man's  will  is  free  because  he  acts  as 
he  is  acted  upon.  This,  it  must  be  acknowledged,  is 
pretty  nearly  the  freedom  of  the  magnetic  needle. 

Now,  in  mathematical  reasening  there  is  none  of 
this  doubt.  Hence,  its  superiority  as  a  means  of  men- 
tal discipline  over  moral  reasoning.  We  here  begin  with 
axioms  self-evident  to  every  intellect,  and  arrive  at  the 
most  important  conclusions,  and  all  along  the  whole 
ascent,  not  a  doubt  springs  up  in  the  mind  lest  the 
outset  may  have  been  wrong.  We  rise  by  a  series  of 
equations  or  ratios,  which  follow  each  other  necessarily, 
to  the  sublime  discoveries  which  occupied  the  attention 
of  a  Descartes  or  a  Newton.  We  begin  by  measuring 
the  lot  on  which  we  build  our  cottage  and  ascend  till 
we  seat  ourselves  among  the  stars  and  contemplate  the 
mechanism  of  the  heavens. 

Again:  Another  obvious  advantage  of  the  study 
of   Mathematics    is,   that    it    serves   to    check    the 


16 

vagaries  of  the  imagination.  This  is  especially  ben- 
eficial to  the  young  student.  The  imagination,  if 
allowed  to  wander  at  will,  strays  into  forbidden  paths, 
and  often  betrays  its  victim  into  fatal  snares.  We 
have  only  to  compare  the  lives  of  the  Mathematicians 
with  the  lives  of  the  Poets,  and  we  shall  find  that 
while  the  former  have  rarely  forgotten  that  they  were 
men  destined  to  a  high  rank  in  the  scale  of  intellec- 
tual beings,  the  latter  have  sacrificed  at  the  shrine  of 
pleasure  till  life  became  a  burden,  and  have  then 
sought  to  drown  a  troubled  conscience  in  the  intoxica- 
ting bowl. 

We  now  come  to  the  practical  advantages  of  this 
study.  And  here  the  prospect  opens  upon  us  so  vast, 
that  we  are  in  danger  of  being  lost  in  the  boundless 
field.  This  science  is  capable  of  being  applied  to  an 
indefinite  extent.  Whether  in  calculating  the  price  of 
a  yard  of  cloth,  or  the  length  of  a  degree  of  longi- 
tude, whether  in  measuring  the  declivity  of  a  roof  or 
the  height  of  a  mountain,  the  width  of  a  river,  or  the 
distance  of  a  planet,  Mathematics  are  always  brought 
into  requisition.  All  the  interesting  phenomena  in 
Optics  may  be  accounted  for  by  a  simple  law  which 
geometry  enables  us  to  verify.  We  wonder  that  the 
puny  arm  of  man  should  have  been  able  to  build  the 
pyramids.  Let  us  wonder  Vather  at  the  science  which 
enabled  man  to  invent  the  machines  that  raised  those 
massive  blocks  to  such  a  height,  and  laid  them  all  in 
their  proper  places.  Without  this  science,  that  puny 
arm  would  have  been  powerless. 

To  the  daily  applications  which  the  different 
branches  of  Mathematics  are  receiving  we  owe,  chiefly, 


17 

the  various  improvements  in  the  arts  which  distinguish 
the  present  age.  The  engineer  has  spanned  our  rivers 
by  bridges — some  of  solid  masonry,  reposing  securely 
upon  semi-circular  arches,  some  upon  attenuated 
threads  of  steel,  and  in  one  instance,  he  has  crossed  a 
strait  more  than  five  hundred  feet  in  width  by  an  elip- 
tical  tube  of  iron  eighteen  feet  by  thirty,  and  this 
raised  so  high  that  the  largest  ships  pass  beneath  it, 
and  so  strong  that  the  longest  trains  of  cars  pass 
through  it  without  causing  it  to  settle  a  single  half 
inch.*  He  has  caused  his  railroads  to  cross  our 
mountains  or  to  pass  beneath  them,  and  thus  to  make 
neighbors  of  people  remote  from  each  other.  He  has 
connected  distant  waters  by  canals,  thereby  opening 
new  markets  to  the  agriculturist  and  giving  an  increased 
impetus  to  trade.  He  has  launched  the  steamship 
upon  the  ocean,  and  commanded  it,  regardless  of  wind 
or  tide,  to  plough  distant  seas  till  now  undisturbed 
but  by  the  monsters  of  the  deep,  and  has  there  made  a 
highway  for  the  nations.  Communicating,  by  means 
of  a  modification  of  the  telescope,  from  the  top  of  one 
high  mountain  with  a  fellow  laborer  upon  the  top  of 
another  high  mountain  many  leagues  distant,  he  fixes 
the  latitude  and  longitude  of  every  important  point 
upon  our  extended  coast,  and  is  enabled  by  this  means 
to  construct  charts,  which  shall  be  safe  guides  to  the 
mariner.  In  a  more  humble  sphere,  he  lays  out  our 
streets  or  surveys  our  lands,  and  fixes  their  limits. 
And  for  all  these  labors  we  are  indebted  to  formulas 

*  Allusion  is  here  made  to  the  iron  bridge  lately  constructed  over 
the  Menai  Straits  by  Mr.  Stevenson.  This  is,  without  doubt,  the 
greatest  triumph  that  engineering  has  yet  achieved. 


18 

depending  upon  calculations  more  or  less  complicated. 

Architecture,  in  its  more  imposing  displays,  -would 
never  have  had  an  existence  but  for  geometry.  From 
the  staircase  to  the  arch  and  the  dome,  all  is  construct- 
ed according  to  rules  furnished  by  geometry.  All  the 
complicated  varieties  of  machinery,  from  the  watch  to 
the  spinning  jenny  and  the  steam  engine,  depend  upon 
mathematical  proportion. 

But  in  no  art  or  science  have  Mathematics  made 
more  brilliant  triumphs  than  in  astronomy.  Planets 
whose  rays  are  lost  before  they  reach  our  unaided 
vision,  are  brought  near  by  the  aid  of  glasses,  and  not 
only  their  distances,  but  their  motions  are  calculated 
with  precision.  We  have  been  so  long  accustomed  to 
look  in  the  calendar  for  the  phases  of  the  moon,  the 
position  of  the  planets,  the  length  of  the  day,  and  in- 
deed, for  all  the  information  we  desire  with  regard  to 
the  heavenly  bodies,  that  we  neither  appreciate  the 
labors  of  those  men  to  whose  indefatigable  researches 
we  are  indebted  for  all  we  know,  nor  do  we  place  an  ad- 
equate estimate  upon  our  knowledge  as  compared  with 
that  of  the  Ancients  who  considered  the  earth  as  a  flat 
surface  of  indefinite  extent,  and  the  sun,  moon  and 
stars  as  pretty  nearly  equidistant  from  the  earth,  around 
which  they  were  supposed  daily  to  revolve.  It  is 
only  three  centuries  since  Tycho  Brahe,  a  Danish  as- 
tronomer who  enriched  the  science  by  the  number  and 
exactness  of  his  observations,  maintained  this  ancient 
hypothesis  in  opposition  to  Copernicus ;  and  it  was  not 
till  Newton  appeared  and,  by  the  light  of  his  power- 
ful mind,  scattered  the  clouds  which  had  hitherto 
hung  over  this  subject,  as  the  sun  scatters  the  mists 


19 

of  the  morning,  that  all  the  civilized  world  adopted  the 
system  of  Copernicus.  And  what  is  true  of  the  An- 
cients is  also  true  of  the  less  enlightened  nations  of  our 
day.  In  not  a  few  instances,  our  Christian  missionaries 
have  won  respect  for  the  great  cause  to  which  they 
have  consecrated  their  lives,  by  contrasting  the  accu- 
racy of  our  almanacs  with  the  erroneous  calendars  of 
the  people  of  India.  To  such  exactness  has  this  science 
arrived  with  us,  that  the  astronomer  shall  predict  to  a 
second  the  commencement  of  an  eclipse  one  hundred 
years  in  advance,  and  at  the  precise  point  of  time  in- 
dicated the  sun, 

"In  dim  eclipse,  disastrous  twilight  sheds 
O'er  half  the  nations,  and  with  fear  of  change 
Perplexes  monarchs." 

Truly,  0  God,  "Thou  hast  made  man  a  little  lower 
than  the  angels,  and  hast  crowned  him  with  glory  and 
honor."  ♦ 


14  DAY  USE 

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